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Properties

Label 4-525e2-1.1-c3e2-0-11
Degree 44
Conductor 275625275625
Sign 11
Analytic cond. 959.512959.512
Root an. cond. 5.565605.56560
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 6·3-s + 4-s + 6·6-s + 14·7-s − 9·8-s + 27·9-s − 22·11-s − 6·12-s + 22·13-s − 14·14-s − 47·16-s − 116·17-s − 27·18-s + 102·19-s − 84·21-s + 22·22-s − 260·23-s + 54·24-s − 22·26-s − 108·27-s + 14·28-s − 196·29-s + 150·31-s + 103·32-s + 132·33-s + 116·34-s + ⋯
L(s)  = 1  − 0.353·2-s − 1.15·3-s + 1/8·4-s + 0.408·6-s + 0.755·7-s − 0.397·8-s + 9-s − 0.603·11-s − 0.144·12-s + 0.469·13-s − 0.267·14-s − 0.734·16-s − 1.65·17-s − 0.353·18-s + 1.23·19-s − 0.872·21-s + 0.213·22-s − 2.35·23-s + 0.459·24-s − 0.165·26-s − 0.769·27-s + 0.0944·28-s − 1.25·29-s + 0.869·31-s + 0.568·32-s + 0.696·33-s + 0.585·34-s + ⋯

Functional equation

Λ(s)=(275625s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(275625s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 275625275625    =    32⋅54⋅723^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 959.512959.512
Root analytic conductor: 5.565605.56560
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 275625, ( :3/2,3/2), 1)(4,\ 275625,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3C1C_1 (1+pT)2 ( 1 + p T )^{2}
5 1 1
7C1C_1 (1−pT)2 ( 1 - p T )^{2}
good2D4D_{4} 1+T+p3T3+p6T4 1 + T + p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+2pT+2718T2+2p4T3+p6T4 1 + 2 p T + 2718 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4}
13D4D_{4} 1−22T+4450T2−22p3T3+p6T4 1 - 22 T + 4450 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+116T+9030T2+116p3T3+p6T4 1 + 116 T + 9030 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−102T+13134T2−102p3T3+p6T4 1 - 102 T + 13134 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+260T+34734T2+260p3T3+p6T4 1 + 260 T + 34734 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+196T+20942T2+196p3T3+p6T4 1 + 196 T + 20942 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−150T+36542T2−150p3T3+p6T4 1 - 150 T + 36542 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−96T+82550T2−96p3T3+p6T4 1 - 96 T + 82550 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+176T−16914T2+176p3T3+p6T4 1 + 176 T - 16914 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−8pT+171958T2−8p4T3+p6T4 1 - 8 p T + 171958 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4}
47D4D_{4} 1+560T+248606T2+560p3T3+p6T4 1 + 560 T + 248606 T^{2} + 560 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+326T+204138T2+326p3T3+p6T4 1 + 326 T + 204138 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+844T+474182T2+844p3T3+p6T4 1 + 844 T + 474182 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+204T+455006T2+204p3T3+p6T4 1 + 204 T + 455006 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−104T+537670T2−104p3T3+p6T4 1 - 104 T + 537670 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−1670T+1384382T2−1670p3T3+p6T4 1 - 1670 T + 1384382 T^{2} - 1670 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−386T+152218T2−386p3T3+p6T4 1 - 386 T + 152218 T^{2} - 386 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+888T+1007454T2+888p3T3+p6T4 1 + 888 T + 1007454 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+928T+600710T2+928p3T3+p6T4 1 + 928 T + 600710 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−588T+1495334T2−588p3T3+p6T4 1 - 588 T + 1495334 T^{2} - 588 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+522T+291282T2+522p3T3+p6T4 1 + 522 T + 291282 T^{2} + 522 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.22830229563661867924912411589, −9.866617179104076557473098780024, −9.292910589293167707581866849496, −9.151774616763100073637836451075, −8.130216543569479799823416786368, −8.117656687226005390987407264881, −7.65155649499516237211362880703, −6.84290374163458097002565157603, −6.58582852725826489533600907087, −6.09193953812086260274418914693, −5.56991007286448801804177232082, −5.15502211710477551068967189687, −4.40949664216371750983075889168, −4.28137017897688057816930304822, −3.39507048704496452027599025816, −2.46535011381250003962058324278, −1.92262791641916818510829055902, −1.22479267346348945102799469931, 0, 0, 1.22479267346348945102799469931, 1.92262791641916818510829055902, 2.46535011381250003962058324278, 3.39507048704496452027599025816, 4.28137017897688057816930304822, 4.40949664216371750983075889168, 5.15502211710477551068967189687, 5.56991007286448801804177232082, 6.09193953812086260274418914693, 6.58582852725826489533600907087, 6.84290374163458097002565157603, 7.65155649499516237211362880703, 8.117656687226005390987407264881, 8.130216543569479799823416786368, 9.151774616763100073637836451075, 9.292910589293167707581866849496, 9.866617179104076557473098780024, 10.22830229563661867924912411589

Graph of the ZZ-function along the critical line